Near-Wall Coherent Structures: History, Identification and Detection - Transport and Coherent Structures in Wall Turbulence - page 162

The analysis given by Chong et al. [CHO 90] for an
incompressible flow is illustrated in Figure 3.19, which also
shows the streamlines seen by an observer moving locally
with the flow. For example, consider the topology at

Δ> 0

and . Both the complex conjugate eigenvalues have a
negative real part. They indicate the presence of a stable
focal point where the trajectories should converge toward the
origin. On the other hand, the real eigenvalue is positive,
which implies that the trajectories move away from the
origin along the axis and stretch the streamlines. The same
kind of reasoning can be applied to the topological structures
in Figure 3.19. In Figure 3.20, we schematically show the
streamlines corresponding to the focal critical point and
node/saddle topologies. The orientation of the trajectories 8
depends on the sign of the invariant

R < 0

.

R

It must be pointed out that the invariants , and in
the tensor remain constant in the presence of a non-
uniform translational motion. This point is important.
Indeed, even if the observer is moving (without rotation) at a
velocity different from the local velocity and consequently
seeing a system of streamlines, which differs from that
shown in Figure 3.20, the local topology remains unchanged,
regardless of the coordinate system chosen. Galilean
invariance implies that the criterion remains the same even
with a coordinate change of the type

R

P Q

J

, where is

y = Tx + a t

T

an orthogonal eigentensor with

() = 1

, and where is a

det T

a

constant velocity vector.

Let us look again at expression [3.29] of the invariant Q

for an incompressible flow

1
2

( )

[3.33] Q = A ij A ij − S ij S ij

8 In the sense of a dynamical system.